mcm-rb Benchmarking Suite

• mcm-rb Benchmarking Suite is a set of randomized benchmarking protocols that quantifies measurement-induced errors and crosstalk in quantum systems.
• It employs variants like mcm-rb, delay-rb, and mcm-rep to isolate control and ancilla error dynamics via survival probability decay measurements.
• The scalable framework integrates process tomography to provide actionable insights for improving the fidelity of fault-tolerant superconducting quantum hardware.

The mcm-rb Benchmarking Suite refers specifically to a family of randomized benchmarking (RB) protocols for characterizing the effect of mid-circuit measurements and associated crosstalk in quantum information processing systems. Built on foundational RB techniques, mcm-rb enables experimentalists to quantify measurement-induced, spectator-qubit, and two-qubit error processes arising during mid-circuit measurements, with particular focus on contemporary superconducting quantum hardware (Govia et al., 2022).

1. Theoretical Foundation and Protocol Overview

mcm-rb extends the randomized benchmarking paradigm, which uses Clifford group twirling to depolarize errors and extract decay constants corresponding to average gate or channel infidelities. In the mcm-rb protocol, the Clifford twirl is confined to the "control" qubit, while an "ancilla" qubit is subject to explicit mid-circuit measurements without additional twirling. This structure is motivated by the experimental constraint that implementing real-time, measurement-conditioned Cliffords is typically not feasible.

The core logic of mcm-rb is to measure the exponential decay of a survival probability (ground-state probability) as a function of sequence length $N$. Sequences are constructed by interleaving single-qubit Clifford operations on the control with either mid-circuit measurements on the ancilla or corresponding idle delays. Multiple sequence types allow the isolation of measurement-induced and spectator errors by differential analysis.

2. Experimental Procedures and Protocol Variants

mcm-rb consists of three core experimental variants, distinguished by the placement and function of measurement or delay within the sequence:

Protocol	Control (C) Operations	Ancilla (A) Operations	Purpose
mcm-rb	Random Cliffords, measure	Mid-circuit measurement (A, each)	Assess control error with measurement
delay-rb	Random Cliffords, idle	Idle delay (duration $t_m$)	Reference for control error (no measurement)
mcm-rep	Idle	Repeated measurement, idle delays	Isolate errors acting solely on the ancilla

For each $N$, one collects statistics over $M$ (e.g., 40–60) random sequences with a large number of shots (e.g., 1024 per sequence). Mid-circuit measurement outcomes themselves are discarded; only the final readout matters.

Experimentally, the routine involves:

1. Initializing both qubits in $\lvert0\rangle$.
2. For each of $N$ cycles: apply a random Clifford to the control; execute either measurement or delay on the ancilla according to protocol.
3. Terminate with an inverting Clifford on the control. Final measurement of both qubits.
4. Repeat for the required number of random sequences and sequence lengths.

This approach is scalable to larger systems by applying simultaneous random Clifford sequences on multiple controls and repeated interleaved ancilla measurements.

3. Analytical Model and Error Quantification

The survival probability on a qubit $q$ in protocol $\nu$ ($q\in\{\text{c},\text{a}\}$, $\nu\in\{\text{rb},\text{del},\text{rep}\}$) is modeled as:

$S_q^\nu(N) = A_q^\nu (\alpha_q^\nu)^N + B_q^\nu$

where $A$ and $B$ absorb state-preparation and measurement (SPAM) errors, and $\alpha_q^\nu$ is the decay parameter.

For single-qubit depolarizing channels, the error per Clifford (EPC) or error per measurement (EPM) is:

$$\operatorname{EPC}_q^\nu = \frac{1 - \alpha_c^\nu}{2}, \quad \operatorname{EPM}_a^\nu = \frac{1 - \alpha_a^\nu}{2}$$

The enhanced, measurement-induced error rate per measurement on the control qubit is extracted by comparing decay parameters:

$$\epsilon_\mathrm{IRM} = \frac{1 - (\alpha_c^{\text{rb}} / \alpha_c^{\text{del}})}{2}$$

Additional error classification is enabled by comparing $$\{\epsilon_c^\text{rb},\epsilon_c^\text{del},\epsilon_c^\text{rep}\}$$ and $$\{\epsilon_a^\text{rb},\epsilon_a^\text{del},\epsilon_a^\text{rep}\}$$, probing non-QND errors, measurement-induced spectator errors, and two-qubit processes.

Common error mechanisms include cross-measurement dephasing ($p_m/3$ infidelity per measurement on the control), Stark shifts ($1-\mathcal{F} = [1-\cos(2\theta)]/3$ per measurement), and resonant two-qubit collisions modeled via $$H = (\Delta/2)Z_a + J(\sigma_a^- \sigma_c^+ + \mathrm{h.c.})$$.

4. Implementation and Calibration Details

On superconducting platforms such as IBM Falcon:

• Clifford gate time $t_g \sim 30$ ns.
• Mid-circuit measurement $t_m \sim 0.7$–$1.0$ μs.
• Relaxation times $T_1 \sim 300$–$400$ μs, $T_2 \sim 200$–$300$ μs.
• $N_\text{max}=150$ cycles, $M=40$ repetitions, 1024 shots per sequence.
• Simultaneous benchmarking of up to 17 qubits (1 ancilla with 2–3 controls per group).

Circuits are synthesized for each specified ancilla-control configuration and experimental data is acquired in parallel to maximize throughput and probe crosstalk.

5. Tomographic Extensions and Error Mechanism Diagnosis

Beyond survival curve fitting, mcm-rb incorporates mid-circuit process tomography to determine the physical nature of observed errors. By constructing the Pauli Transfer Matrix (PTM) $$R_{ij} = \operatorname{Tr}[P_i \mathcal{E}(P_j)] / 2$$, experimenters can diagnose:

• Ideal block-diagonal PTM: no cross-qubit influence; ancilla operators project to measurement subspace.
• Off-diagonal PTM elements: coherent crosstalk, Stark shifts, or collision-induced entangling dynamics.

Comparison of empirically reconstructed PTMs to analytical error models confirms and quantifies the dominant error processes.

6. Scalability and Multi-Qubit Generalization

mcm-rb is directly extensible to benchmarking multiple device regions or concurrent ancilla-control pairs. By using independent random Clifford streams per control and parallel interleaved measurements per ancilla, one can efficiently extract error rates under scalable circuit loads. Infidelities are additive in the small error regime:

• Ancilla: total EPM scales linearly with the number of measurements.
• Control: total EPC comprises intrinsic gate infidelity and the sum of measurement-induced errors across all measured ancillas.

This modularity enables simultaneous characterization of spatially resolved measurement-induced error phenomena, crosstalk, and performance bottlenecks in large-scale quantum processors.

7. Significance in Quantum Device Characterization

The mcm-rb suite provides a rigorous, scalable, and experimentally feasible framework for quantifying the operational fidelity of mid-circuit measurements and their impact on both targeted and spectator subsystems (Govia et al., 2022). Its experimental design and analytical modeling rigorously distinguish measurement-induced infidelities from general gate processes, thus delivering actionable benchmarks for quantum hardware improvements. By integrating standard randomized benchmarking techniques with explicit treatment of mid-circuit measurements and cross-talk diagnostics, mcm-rb forms a key protocol for validating platforms aimed at fault-tolerant quantum computation and measurement-based quantum information processing.